Question

Упростите:

$\frac{1}{ctg\alpha}+\frac{cos\alpha}{1+sin\alpha}$

Ответ $\frac{1}{cos\alpha}}$

Answer 1

Легко:
1/ctga=1/(cosa/sina)=sina/cosa.

Sina/cosa+cosa/(1+sina)=((sina*(1+sina)+cos^2(a))/(cosa+cosa*sina)=(sina+sin^2(a)+cos^2(a))/(cosa*(1+sina)=(sina+1)/(cosa*(1+sina))=1/cosa.

Answer 2

$\frac{1}{ctg\alpha} + \frac{cos\alpha}{1+sin\alpha} = \\ =tg\alpha + \frac{cos\alpha}{1+sin\alpha} = \\ =\frac{sin\alpha}{cos\alpha} + \frac{cos\alpha}{1+sin\alpha} = \\ =\frac{sin\alpha+sin^2\alpha + cos^2\alpha}{cos\alpha(1+sin\alpha)}=\\ = \frac{sin\alpha+sin^2\alpha + cos^2\alpha}{cos\alpha(1+sin\alpha)} =\\ = \frac{1+sin\alpha}{cos\alpha(1+sin\alpha)} = \frac{1}{cos\alpha}$